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WARM UP. Given : AB || CD AE BE. Prove : 3 4. E. 3. 4. C. D. 1. 2. A. B. Statements Reasons. 1. AB || CD. 1. Given. 2. If lines are parallel, then corresponding angles are congruent. 2. 1 3, 2 4. 3. AE BE.
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WARM UP Given: AB || CD AE BE Prove: 3 4 E 3 4 C D 1 2 A B
Statements Reasons 1. AB || CD 1. Given 2. If lines are parallel, then corresponding angles are congruent. 2. 1 3, 2 4 3. AE BE 3. Given 4. If two sides of a triangle are congruent, then the angles opposite those sides are congruent. 4. 1 2 5. 2 3 5. Substitution 6. 3 4 6. Substitution
Section 4-5 AAS, HL Theorems & Proofs
AAS Theorem: If two angles and a non-included side of one triangle are congruent to the corresponding parts of another triangle, then the triangles are congruent. A ABC ___ DEF D C B F E
C Example 1) ABC ___ DBC AAS Theorem D A B Example 2) A D ABC ___ EDC C AAS Theorem B E
HL Theorem: If the hypotenuse and a leg of one right triangle are congruent to corresponding parts of another right triangle, then the triangles are congruent. E C A F B D ABC ___ DFE
B Example 1) ABC ___ DBC HL Theorem A D C Example 2) C D ABC ___ DCB HL Theorem A B
PROOF EXAMPLE 1: V Given: 1 2 CD bisects VCW Prove: DV DW 1 3 D C 4 2 W Statements Reasons 1. CD bisects VCW 1. Given 2. 3 4 2. Def. of Angle Bisector 3. 1 2 6. DV DW 3. Given 4. CD CD 4. Reflexive Property 5. ΔCVD ΔCWD 5. AAS Theorem 6. CPCTC
PROOF EXAMPLE 2: X Given: W and Y are right angles WX YX Prove: WZ YZ Y W Z Statements Reasons 1. W and Y are right angles 1. Given 2. ΔXWZ andΔXYZ are right triangles. 2. Def. of Right Triangle 3. WX YX 6. WZ YZ 3. Given 4. XZ XZ 4. Reflexive Property 5. ΔXWZ ΔXYZ 5. HL Theorem 6. CPCTC
PROOF EXAMPLE 3: Given: KL LA; KJ JA; AK bisects LAJ Prove: LK JK L 1 K A 2 J
Statements Reasons 1. Given 1. KL LA; KJ JA 2. Land J are right angles 2. Def. of Perpendicular Lines 3. mL = 90; mJ = 90 3. Def. of Right Angles 4. mL= mJ; L J 4. Substitution • AK bisects LAJ 5. Given 6. 1 2 6. Def. of Angle Bisector 7. KA KA 7. Reflexive Property 8. ΔLKA ΔJKA 8. AAS Theorem 9. LK JK 9. CPCTC